Why does convolution effect persist despite taking infinite projections?

[EeePC1005PR]

I appreciate the basics of forward and back projection in CT, using forward projected results at different angles, work backwards to estimate the values of original voxels.

I also see there is a fuzzy rim around the voxel when we take numerous projections, with 1/r functional dependence on distance r, and we can apply convolution/ramp filter to pre-subtract that effect off. I got the filter bit. Just pre-empting the effect and taking it off in advance.

My question is: why is there this fuzzy rim in the first place?

Suppose we take infinite projections, shouldn't we then have enough data to accurately describe each single voxel (so accurate we don't even get the fuzzy rim)? But that's not the case. In truth even if we took infinite projections, that fuzzy rim will still be there. Why?

Sure at 4:50 of the clip I see spokes, which become shorter and blunter as we increase the number of projections. One can say "see, as you increase the projections, those spokes flatten out and fuse to become the halo". But that still doesn't make sense. The spokes are just overlapping ADJACENT projections. Our back projection calculation is only concerned with overlap of ALL projections, not just adjacent projections. Overlapping bits of adjacent projections are automatically ejected from our calculations, so those spokes shouldn't have anything to do with the halo effect.

https://www.youtube.com/watch?v=8V2QBD8nh_s

Graph at 5:10 suggests no matter how many projections we increase to, there will always be that fall off rim. Why?

Hope my question is clear enough. I can appreciate there's inaccurate rim with just few projections, but why persistent halo even at infinite projections? Just seeking kind explanation for this mathematical/physical phenomenon in simple English please.

Many thanks.

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[Shifty B]

This is part of an inherent limitation in back projection itself, related to the fact that the data is being acquired radially and then reconstructed in a rectangular space. There is a more detailed explanation here:

http://www.dspguide.com/ch25/5.htm

This blurring is not only related to the number of projections but also related to the number of detectors (i.e. the resolution of the detector). By increasing the resolution of the detector, the blurring effect is also reduced. If the object is significantly larger than the detector, then the blurring effect becomes smaller. If you have an infinite number of projections and detectors, then there is no blurring (theoretically of course).

 

[EeePC1005PR]

This is part of an inherent limitation in back projection itself, related to the fact that the data is being acquired radially and then reconstructed in a rectangular space. There is a more detailed explanation here:

http://www.dspguide.com/ch25/5.htm

This blurring is not only related to the number of projections but also related to the number of detectors (i.e. the resolution of the detector). By increasing the resolution of the detector, the blurring effect is also reduced. If the object is significantly larger than the detector, then the blurring effect becomes smaller. If you have an infinite number of projections and detectors, then there is no blurring (theoretically of course).

Hey thanks for the link and explanation Shifty.

My initial gut instinct was even with infinite projections, the halo will still be there. However the website you provided actually says "the image produced by filtered backprojection is identical to the 'correct' image when there are an infinite number of views and an infinite number of points per view." So I guess that does imply infinite projections won't create the fuzzy rim problem.

I still got the question though: how exactly does the fuzzy rim come about?

Looking at figure 25-16, probably the way those signals have round tops are the cause. So why are those signals rounded? Why aren't they square waves with flat tops, then probably we won't have convolution problem.

 

[Shifty B]

Looking at figure 25-16, probably the way those signals have round tops are the cause. So why are those signals rounded? Why aren't they square waves with flat tops, then probably we won't have convolution problem.

X-rays are attenuated by the cumulative amount of matter they pass through. If you go in a straight line through the center, that is passing through more than a straight line, say half way along the radius towards either side.

You would get that same problem even with a square, though, because as you rotate the distance traversed varies based on how far you are from the center.